In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration"
Suppose $U$ is an open subset of the complex plane ${\bf C}$, $f : U \rightarrow {\bf C}$ is a holomorphic function and the closed disk $D = \{ z : | |z - z_0| \leq r\}$ is completely contained in $U$. Let $\gamma$ be the circle forming the boundary of $D$. Then for every $a$ in the interior of $D$ : $$f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\ dz$$ where the contour integral is taken counter-clockwise.
In particular $f$ is actually infinitely differentiable, with $$ f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\ dz$$ This formula is sometimes referred to as Cauchy's differentiation formula.
